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Abstract
We show that every finitely generated group G with an element of order at least (5rank(G))12 admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least (2rank(G))36, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi vertextransitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.
Original language  English 

Pages (fromto)  11171146 
Number of pages  30 
Journal  Journal of Algebraic Combinatorics 
Volume  53 
Issue number  4 
DOIs  
Publication status  Published  Jun 2021 
Externally published  Yes 
Keywords
 Automorphisms of graphs
 Cayley graph
 Covering
 DRR
 Digraphical rigid representation
 GRR
 Graphical rigid representation
 ORR
 Oriented rigid representation
 Regular automorphism group
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Dive into the research topics of 'Cayley graphs with few automorphisms'. Together they form a unique fingerprint.Activities
 1 Invited talk

Rigidity of Cayley graphs
PaulHenry Leemann (Speaker)
10 Nov 2023Activity: Talk or presentation › Invited talk